1 Answer
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What you refer to as 'Hash 160 Address' and which appear in this example under the field 'Hash 160' (e.g. the hex number a6c70c4a88205065c1d33b17c156137fa8c736c1) is a 20 bytes hash of the public key (specifically, it is the ripemd160 hash of the sha256 hash of the public key). These 20 bytes are essentially the same thing as what you call the 'Public Bitcoin address' (e.g. 1GCqcweZTwwe1w2RMWwwQMKSFgQzgFZQUb in our example). This bitcoin address is a 25 bytes number (usually displayed in Base58) which is linked to the hash 160 as follows:

address = [Version Byte (1)][Hash 160 (20)][Checksum (4)]

The version byte is simply 0x00 for the main bitcoin network and the 4 checksum bytes are simply the first 4 bytes of the double sha256 hash of the 21 bytes [Version Byte][Hash 160]

Thanks Sven, that's clear and understand the differences now. Can you tell me why both are needed? Also, are both transmitted in the transaction or is one just being calculated and displayed by the website I took the screenshot from?
– monkiaJan 19 '17 at 18:44

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The two things represent the same number, but they are both useful because on the one hand dealing with a bitcoin address gives us the security of having an embedded 4 bytes checksum (so we don't transfer money by mistake to something that isn't a valid address), and on the other hand, dealing with the Hash 160 only (20 bytes) is more efficient for the purpose of encoding a transaction. If you look at the actual bytes of a transaction encoding (with this address as output), you'll see that the 20 bytes of the hash 160 appear near the end (e.g. bitcoin-cli getrawtransaction <txid>)
– Sven WilliamsonJan 19 '17 at 21:54

Note that the leading version byte added to the address is also useful to prevent someone mistakenly transferring real money while attempting to test functionality on a bitcoin testing network.
– Sven WilliamsonJan 19 '17 at 22:11